Anthony R. Lupo Department of Soil, Environmental, and
Atmospheric Sciences 302 E ABNR Building University of Missouri
Columbia, MO 65211
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Some popular images..
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Any attempt at weather forecasting is immoral and damaging to
the character of a meteorologist - Max Margules (1904) (1856 1920)
Margules work forms the foundation of modern Energetics
analysis.
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Chaotic or non-linear dynamics Is perhaps one of the most
important discovery or way of relating to and/or describing natural
systems in the 20 th century! Chaos and order are opposites in the
Greek language - like good versus evil. Important in the sense that
well describe the behavior of non-linear systems!
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Physical systems can be classified as: Deterministic laws of
motion are known and orderly (future can be directly determined
from past) Stochastic / random no laws of motion, we can only use
probability to predict the location of parcels, we cannot predict
future states of the system without statistics. Only give
probabilities!
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Chaotic systems We know the laws of motion, but these systems
exhibit random behavior, due to non-linear mechanisms. Their
behavior may be irregular, and may be described statistically. E.
Lorenz and B. Saltzman Chaos is order without periodicity.
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Classifying linear systems If I have a linear set of equations
represented as: (1) And b is the vector to be determined. Well
assume the solutions are non-trivial. Q: What does that mean again
for b? A: b is not 0!
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Eigenvalues are a special set of scalars associated with a
linear system of equations (i.e., a matrix equation) that are
sometimes also known as characteristic roots (source: Mathworld) (
) Eigenvectors are a special set of vectors associated with a
linear system of equations (i.e., a matrix equation) that are
sometimes also known as characteristic vectors (vector b)
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Thus we can easily solve this problem since we can substitute
this into the equations (1) from before and we get: Solve, and so,
now the general solution is: Values of c are constants of course.
The vectors b1 and b2 are called eigenvectors of the eigenvalues 1
and 2.
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Particular Solution:
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One Dimensional Non-linear dynamics We will examine this
because it provides a nice basis for learning the topic and then
applying to higher dimensional systems. However, this can provide
useful analysis of atmospheric systems as well (time series
analysis). Bengtssen (1985) Tellus Blocking. Federov et al. (2003)
BAMS for El Nino. Mokhov et al. (1998, 2000, 2004). Mokhov et al.
(2004) for El Nino via SSTs (see also Mokhov and Smirnov, 2006),
but also for temperatures in the stratosphere. Lupo et al (2006)
temperature and precip records. Lupo and Kunz (2005), and Hussain
et al. (2007) height fields, blocking.
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First order dynamic system: (Leibnitz notation is x dot?) If x
is a real function, then the first derivative will represent a(n)
(imaginary) flow or velocity along the x axis. Thus, we will plot x
versus x dot
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Draw: Then, the sign of f(x) determines the sign of the one
dimensional phase velocity. Flow to the right (left): f(x) > 0
(f(x) < 0)
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Two Dimensional Non-linear dynamics Note here that each
equation has an x and a y in it. Thus, the first deriviatives of x
and y, depend on x and y. This is an example of non-linearity. What
if in the first equation Ax was a constant? What kind of function
would we have? Solutions to this are trajectories moving in the
(x,y) phase plane.
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Coupled set: If the set of equations above are functions of x
and y, or f(x,y). Uncoupled set: If the set of equations above are
functions of x and y separately.
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Definitions Bifurcation point: In a dynamical system, a
bifurcation is a period doubling, quadrupling, etc., that
accompanies the onset of chaos. It represents the sudden appearance
of a qualitatively different solution for a nonlinear system as
some parameter is varied.
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Example: pitchfork bifurcation (subcritical) Solution has three
roots, x=0, x 2 = r
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The devil is in the details?............
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An attractor is a set of states (points in the phase space),
invariant under the dynamics, towards which neighboring states in a
given basin of attraction asymptotically approach in the course of
dynamic evolution. An attractor is defined as the smallest unit
which cannot be itself decomposed into two or more attractors with
distinct basins of attraction
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How we see it.
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Mathematics looks at Equation of Motion (NS) is space such
that: Closed or compact space such that boundaries are closed and
that within the space divergence = 0 Complete set div = 0 and all
the interesting sequences of vectors in space, the support space
solutions are zero.
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Ok, lets look at a simple harmonic oscillator (pendulum): Where
m = mass and k = Hookes constant. When we divide through by mass,
we get a Sturm Liouville type equation.
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One way to solve this is to make the problem self adjoint or to
set up a couplet of first order equations like so let:
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Then divide these two equations by each other to get: What kind
of figure is this?
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A set of ellipses in the phase space.
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Here it is convenient that the origin is the center! At the
center, the flow is still, and since the first derivative of x is
positive, we consider the flow to be anticyclonic (NH) clockwise
around the origin. The eigenvalues are: Now as the flow does not
approach or repel from the center, we can classify this as
neutrally stable.
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Thus, the system behaves well close to certain fixed points,
which are at least neutrally stable. System is forever predictable
in a dynamic sense, and well behaved. we could move to an area
where the behavior changes, a bifurcation point which is called a
separatrix. Beyond this, system is unpredictable, or less so, and
can only use statistical methods. Its unstable!
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Hopfs Bifurcation: Hopf (1942) demonstrated that systems of
non-linear differential equations (of higher order that 2) can have
peculiar behavior. These type of systems can change behavior from
one type of behavior (e.g., stable spiral to a stable limit cycle),
this type is a supercritical Hopf bifurcation.
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Hopfs Bifurcation: Subcritical Hopf Bifurcations have a very
different behavior and these we will explore in connection with
Lorenzs equations, which describe the atmospheres behavior in a
simplistic way. With this type of behavior, the trajectories can
jump to another attractor which may be a fixed point, limit cycle,
or a strange attractor (chaotic attractor occurs in 3 D only!)
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Example of an elliptic equation in meteorology:
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Taken from Lupo et al. 2001 (MWR)
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Ok, lets modify the equation above: d is now the damping
constant, so lets damp (add energy to) this expression d > 0
(d< 0). Then the oscillator loses (gains) energy and the
determinant of the quadratic solution is also less (greater) than
zero! So trajectories spin toward (away from) the center. This is
a(n) (un)stable spiral.
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Example: forced Pendulum (J. Hansen)
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Another Example: behavior of a temperature series for Des
Moines, IA (taken from Birk et. al. 2010)
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Another Example: behavior of 500 hPa heights in the N. Hemi.
(taken from Lupo et. al. 2007, Izvestia)
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Sensitive Dependence on Initial Conditions (SDIC not a federal
program ). Start with the simple system : A iterative-type equation
used often to demonstrate population dynamics: Experiment with k =
0.5, 1.0, 1.5, 1.6, 1.7, 2.0
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For each, use the following x n and graph side-by- side to
compare the behavior of the system. X n = -0.5, X n = -0.50001 Try
to find: period 2 attractor or attracting point: behavior1
behavior2 behavior 1 behavior2, and a period 4 attractor. Period 2
behaves like the large-scale flow?
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Examine the initial conditions. One can be taken to be a
measurement and the other, a deviation or error, whether its
generated or real. Its a point in the ball-park of the original.
Asside: Heisenbergs Uncertainty Principle All measurements are
subject to a certain level of uncertainty.
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X = 0.5X = 0.50001
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Whats the diff?
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The differences that emerge illustrate the concept of Sensitive
Dependence on Initial Conditions (SDIC). This is an important
concept in Dynamic systems. This is also the concept behind
ENSEMBLE FORECASTING! Toth, Z., and E. Kalnay, 1993: Ensemble
forecasting at NCEP: The generation of perturbations. Bull. Amer.
Meteor. Soc., 74, 2317 2330. Toth, Z., and E. Kalnay, 1997:
Ensemble forecasting at NCEP and the breeding method. Mon. Wea.
Rev., 125, 3297 3319. Tracton, M.S., and E. Kalnay, 1993: Ensemble
forecasting at the National Meteorological Center: Practical
Aspects. Wea. and Forecasting, 8, 379 39
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The basic laws of geophysical fluid dynamics describe fluid
motions, they are a highly non-linear set of differentials and/or
differential equations. e.g., Given the proper set of initial
and/or boundary conditions, perfect resolution, infinite computer
power, and precise measurements, all future states of the
atmosphere can be predicted forever!
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Given that this is not the case, these equations have an
infinite set of solutions, thus anything in the phase space is
possible. In spite of this, the same solutions appear time and time
again!
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Note: We will define Degrees of Freedom here this will mean the
number of coordinates in the phase space. Advances in this area
have involved taking expressions with an infinite number of degrees
of freedom and replacing them with expressions of finite degrees of
freedom. For the equation of motion, whether we talk about math or
meteorology, we usually examine the N-S equations in 2- D sense.
Mathematically, this is one of the Million dollar problems to solve
in 3-d (no uniqueness of solutions!).
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Chaotic Systems: 1. A system that displays SDIC. 2. Possesses
Fractal dimensionality
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Fractal geometry self similar Norwegian ModelL. Lemon
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Fractals: Fractal geometry was developed by Benoit Mandelbrot
(1983) in his book the Fractal Geometry of Nature. Fractal comes
from Fractus broken and irregular. Fractals are precisely a
defining characteristic of the strange attractor and distinguishes
these from familiar attractors.
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3. Dissipative system: Lyapunov Exponents - defined as the
average rates of exponential divergence or convergence of nearby
trajectories. They are also in a very real sense, they provide a
quantitative measure of SDIC. Lets introduce the concept using the
simplest type of differential equation.
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Simple differential equation: with the solution as:
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Thus, after some large time interval t, the distance (t)
between two points initially separated by (0) is: Thus, the SIGN of
the exponent here is of crucial importance!!!! A positive value for
infers that trajectories separate at an exponential rate, while a
negative value implies convergence as t infinity!
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Well, we can use our simple differential equation to get the
value of the exponent as: So, in the general case of our
differential equation, we can think of a (particular) solution as a
point on the phase space, and the neighboring points as
encompassing an n- dimensional ball of radius (0)! With an increase
in time, the ball will become an ellipsoid in non-uniform flow, and
will continue to deform as time approaches infinity.
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There must be, by definition, as many Lyapunov exponents as
there are dimensions in the phase space. Again, positive values
represent divergence, while negative values indicate convergence of
trajectories, which represent the exponential approach to the
initial state of the Attractor!
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There must also, by definition, be one exponent equal to zero
(which means the solution is unity) or corresponds to the direction
along the trajectory, or the change in relative
divergence/convergence is not exponential. Now, for a dissipative
system, all the trajectories must add up to be negative!
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Lorenz (1960), Tellus: First Low Order Model (LOM) in
meteorology, derived using Galerkin methods, which approximate
solutions using finite series. (e.g. Haltiner and Williams,
1980).
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Lorenz (1960) Tellus
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Lorenz (1963), J. Atmos. Sci., 20, 130 - 142 Investigated
Rayleigh Bernard (RB) convection, a classical problem in physics.
We need to scale the primitive equations (use Boussinesq approx),
then use Galerkin Techniques again.
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Lorenz (1963) solution:
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Lorenz (1963) then using the initial conditions: = 10.0, b =
8/3, r = 28.0, and a non-dimensional time step of 0.0005. Then
using 50 lines of FORTRAN code, and the leapfrog method, we can
produce:
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The Butterfly
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We cannot solve Lorenzs (1963) LOM unless we examine steady
state conditions; that is dx/dt, dy/dt, and dz/dt all equal zero.
The trivial solution x = y = z = 0, is the state of no
convection.
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But, if we solve the equations, we get some interesting roots;
(0 < r < 1).
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But when r > 1, we get convection and chaotic motions:
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Predictability: SDIC in the flow exists in set A if there
exists error > 0, such that for any and any neighborhood U of x,
there exists and t > 0. such that
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In plane English: there will always be SDIC in a system (its
intrinsic to many systems). Possible outcomes are larger than the
error in specifying correct state! SDIC means that trajectories are
unpredictable, even if the dynamics of a system are well-known
(deterministic).
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Thus, if you wish to compute trajectories of X in a system
displaying SDIC, after some time t, you will accumulate error in
the prediction regardless of increases in computing power! There is
always resolution and measurement error to contend with as well.
This will further muddy the waters.
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Singular Values and Vectors Is the factor by which initial
error will grow for infinitesimal errors over a finite time at a
particular location (singular vectors, as the name implies, give
the direction). Can be numerically estimated using linear theory.
Singular values/vectors are dependent upon the choice of norm; they
are critically state dependent.
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Thus, after some large time interval t, the distance (t)
between two points initially separated by (0) is (from slide 48 and
49):
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Thus, if the error doubles, or the ratio between one trajectory
and another: and the time to accomplish this is:
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This is the basis for stating that the predictability of
various phenomena is about the size of its growth period. For
extratropical cyclones this is approximately 0.5 3 days. For the
planetary scale, the time period is roughly 10 14 days (evolution
of large-scale troughs and ridges).
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This is why we say that 10 14 days is the of time is the limit
of dynamic weather prediction. In atmospheric science, we know that
this is the time period for the evolution of Rossby inertia waves,
which are the result of the very size and rotation rate of the
planet earth! (f = 2 sin )
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Now, the question is, if we know exactly the initial state (is
it possible to know this?) of the atmosphere at some time t, can we
make perfect forecasts? This question is central to the contention
that the atmosphere contains a certain amount of inherent
unpredictability.
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Laplace argued that given the entire and precise state of the
universe at any one instant, the entire cosmos could be predicted
forever and uniquely, by Newtons Laws of motion. He was a firm
believer in determinism.
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But, can we know the exact initial state? Lets revisit
Heisenberg! Exact solutions do exist, so in theory we can find
them. What we can never do even in principle - is specify the exact
initial conditions!
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Measurement error and predictability: If we solve for t (as we
did earlier for error- doubling) : Where h is the sum of the
positive Lyapunov exponents.
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Suppose our uncertainty is at a level of 10 -5, then: Now, lets
improve the accuracy by 5 orders of magnitude, or 10 -10 :
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Then, we should be able to infer that: Or, this increase in
precision only doubles the forecast time. Thus, input error, will
swell very quickly! Should we be pessimistic?
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Not a great return on investment! Pessimistic about our
prospects on forecasting? From a selfish standpoint, no because
this demonstrates that we cannot turn over weather forecasting to
computers. From a scientific standpoint, no as well, because we
just need to realize that forecasting beyond a certain limit at a
certain scale is inevitable. As long as we realize the limitations,
we can make good forecasts.
Slide 80
One beneficial issue has been stimulated for operational
meteorology by Chaos Theory, and that is how do we express
uncertainty in forecasts? Example:
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The End!
Slide 82
Overtime! Fractal Dimension: Were used to integer whole numbers
for dimensionality, but the Fractal can have a dimensionality that
is not a whole number. For example, the Koch Snowflake (1904)
dimension is 1.26.
Slide 83
What? How can you have 1.25 dimensions? But the snowflake fills
up space more efficiently than a smooth curve or line (1-D) and is
less efficient than an area (2-d). So a dimension between one and
two captures this concept.
Slide 84
Example: (Sierpinski Gasket, 1915) Has a Fractal (Hausdorf)
dimension of 1.59
Slide 85
Hausdorf dimension: d = ln(N( )) / [ln(L) ln( )] N( ) = is the
smallest number of cubes (Euclidian shapes) needed to cover the
space. Here it is: 3 n or makes 3 copies of itself with each
iteration.
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The denominator is: ln( L / ) where L = 1 (full space) and is
copy scale factor ((1/2) n length of full space with each
iteration). So we get: d = n ln(3) / n ln (2) = 1.59
